Quantum Mechanics and Path Integrals: Quadratic Actions

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2022, Tutor: Josep Tarón RocaThis work aims to calculate the Feynman propagator of several physical systems governed by quadratic actions by means of the Path Integral approach. Consequently, once the propagator is known for the case of a harmonic potential, a system of coupled oscillators will be studied. Finally, it will be possible to determine the time evolution of a Gaussian wave function when its width is initially modified, as well as the effect induced by a periodic external forc

Relative age of Cordilleran base metal lode and replacement deposits, and high sulfidation Au-(Ag) epithermal mineralization in the Colquijirca mining district, central Peru

At Colquijirca, central Peru, a predominantly dacitic Miocene diatreme-dome complex of 12.4 to 12.7Ma (40Ar/39Ar biotite ages), is spatially related to two distinct mineralization types. Disseminated Au-(Ag) associated with advanced argillic alteration and local vuggy silica typical of high- sulfidation epithermal ores are hosted exclusively within the volcanic center at Marcapunta. A second economically more important mineralization type is characterized as "Cordilleran base metal lode and replacement deposits." These ores are hosted in Mesozoic and Cenozoic carbonate rocks surrounding the diatreme-dome complex and are zoned outward from pyrite-enargite-quartz-alunite to pyrite-chalcopyrite-dickite-kaolinite to pyrite-sphalerite-galena-kaolinite-siderite. Alunite samples related to the Au-(Ag) epithermal ores have been dated by the 40Ar/39Ar method at 11.3-11.6Ma and those from the Cordilleran base metal ores in the northern part of the district (Smelter and Colquijirca) at 10.6-10.8Ma. The significant time gap (~0.5 My) between the ages of the two mineralization types in the Colquijirca district indicates they were formed by different hydrothermal events within the same magmatic cycle. The estimated time interval between the younger mineralization event (base metal mineralization) at ~10.6Ma and the ages of ~12.5Ma obtained on biotites from unmineralized dacitic domes flanking the vicinity of the diatreme vent, suggest a minimum duration of the magmatic-hydrothermal cycle of around 2Ma. This study on the Colquijirca district offers for the first time precise absolute ages indicating that the Cordilleran base metal lode and replacement deposits were formed by a late hydrothermal event in an intrusive-related district, in this case post Au-(Ag) high-sulfidation epithermal mineralizatio

Cenozoic continental arc magmatism and associated mineralization in Ecuador

Most of the economic ore deposits of Ecuador are porphyry-Cu and epithermal style gold deposits associated with Tertiary continental arc magmatism. This study presents major and trace element geochemistry, as well as radiogenic isotope (Pb, Sr) signatures, of continental arc magmatic rocks of Ecuador of Eocene to Late Miocene (~50-9Ma, ELM) and Late Miocene to Recent (~8-0Ma, LMR) ages. The most primitive ELM and LMR rocks analyzed consistently display similar trace element and isotopic signatures suggesting a common origin, most likely an enriched MORB-type mantle. In contrast, major and trace element geochemistry, as well as radiogenic isotope systematics of the whole sets of ELM and LMR samples, indicate strikingly different evolutions between ELM and LMR rocks. The ELM rocks have consistently low Sr/Y, increasing Rb/Sr, and decreasing Eu/Gd with SiO2, suggesting an evolution through plagioclase-dominated fractional crystallization at shallow crustal levels (20km), and most likely at sub-crustal levels (>40-50km). The change in geochemical signatures of Tertiary magmatic rocks of Ecuador from the ELM- to the LMR-type coincides chronologically with the transition from a transpressional to a compressional regime that occurred at ~9Ma and has been attributed by other investigations to the onset of subduction of the aseismic Carnegie ridge. The major districts of porphyry-Cu and epithermal deposits of Ecuador (which have a small size, <<200Mt, when compared to their Central Andean counterparts) are spatially and temporally associated with ELM magmatic rocks. No significant porphyry-Cu and epithermal deposits (except the epithermal high-sulfidation mineralization of Quimsacocha) appear to be associated with Late Miocene-Recent (LMR, ~8-0Ma) magmatic rocks. The apparent "infertility” of LMR magmas seems to be at odds with the association of major porphyry-Cu/epithermal deposits of the Central Andes with magmatic rocks having adakite-type geochemical signatures similar to LMR rocks. The paucity of porphyry-Cu/epithermal deposits associated with LMR rocks might be only apparent and bound to exposure level, or real and bound (among other possibilities) to the lack of development of shallow crustal magmatic chambers since ~9Ma as a result of a prolonged compressional regime in the Ecuadorian crust. More work is needed to understand the actual metallogenic potential of LMR rocks in Ecuado

Within-plate Volcanism in upper Triassic to lower Jurassic Pucara Group carbonates (Central Peru)

The Pucará Group platform carbonates (Upper Triassic - Lower Jurassic) were laid down in northern and central Peru in a NNW-SSE elongated basin (Fig. 1). They represent the first sediments of the Andean cycle, the beginning of which is marked by a Norian transgression (Mégard, 1978). The sedimentary evolution of the Pucará Group can be explained in terms of a large transgressive/regressive second order sequence which consists of predominantly shallow water carbonates including a maximum flooding period represented by ammonite-bearing bituminous calcareous shales. Detailed investigations in the southern part of the basin show that the Pucará Group thickens progressively from west to east in the form of a half-graben (Fig. 2). This can be explained by asymmetrical subsidence during sedimentation such being assisted by contemporaneous faulting along the eastern margin of the basin (permitting rapid subsidence) and a stable hinge zone to the west. Synsedi - mentary tectonics at the eastern edge led to the formation of discrete structural blocks with extreme variations in thickness and facies. It has been suggested that, during burial diagenesis, these faults served as channelways for the basinal brines responsible for MVT-mineralization (Fontboté et al., 1995, Spangenberg, 1995, and Moritz et al., 1996)

Mineral zoning and gold occurrence in the Fortuna skarn mine, Nambija district, Ecuador

The Fortuna oxidized gold skarn deposit is located in the northern part of the Nambija gold district, southern Ecuador. It has been subdivided into four mineralized sites, covering a distance of 1km, which are named from north to south: Cuerpo 3, Mine 1, Mine 2, and Southern Sector. Massive skarn bodies occur in K-Na metasomatized volcanic and volcaniclastic rocks of the Triassic Piuntza unit. They appear to result from selective replacement of volcaniclastic rocks. Very minor presence of bioclast relicts suggests the presence of subordinate limestone. Endoskarn type alteration with development of Na-rich plagioclase, K-feldspar, epidote, actinolite, anhedral pyroxene, and titanite affects a quartz-diorite porphyritic intrusion which crops out below the skarn bodies in Mine 2 and the Southern Sector. Endoskarn alteration in the intrusion grades into a K-feldspar ± biotite ± magnetite assemblage (K-alteration), suggesting that skarn formation is directly related to the quartz-diorite porphyritic intrusion, the latter being probably emplaced between 141 and 146Ma. The massive skarn bodies were subdivided into a dominant brown garnet skarn, a distal green pyroxene-epidote skarn, and two quartz-rich varieties, a blue-green garnet skarn and light green pyroxene-garnet skarn, which occur as patches and small bodies within the former skarn types. The proximal massive brown garnet skarn zone is centered on two 060° trending faults in Mine 2, where the highest gold grades (5-10g/t) were observed. It grades into a distal green pyroxene-epidote skarn zone to the North (Cuerpo 3). Granditic garnet shows iron enrichment from the proximal to the distal zone. Diopsidic pyroxene exhibits iron and manganese enrichment from proximal to distal zones. The retrograde stage is weakly developed and consists mainly of mineral phases filling centimeter-wide veins, vugs, and interstices between garnet and pyroxene grains. The main filling mineral is quartz, followed by K-feldspar, epidote, calcite, and chlorite, with minor sericite, apatite, titanite, hematite, pyrite, chalcopyrite, and gold. Metal and sulfur contents are low at Fortuna, and the highest gold grades coincide with high hematite abundance, which suggests that retrograde stage and gold deposition took place under oxidizing conditions. Fluid inclusions from pyroxene indicate precipitation from high temperature—high to moderate salinity fluids (400 to 460°C and 54- to 13-wt% eq. NaCl), which result probably from boiling of a moderately saline (∼8-wt% eq. NaCl) magmatic fluid. Later cooler (180 to 475°C) and moderate to low saline fluids (1- to 20-wt% eq. NaCl) were trapped in garnet, epidote, and quartz, and are interpreted to be responsible for gold deposition. Chlorite analysis indicates temperature of formation between 300 and 340°C in accordance with fluid inclusion data. It appears, thus, that gold was transported as chloride complexes under oxidizing conditions and was deposited at temperatures around 300°C when transport of chloride complexes as gold carriers is not efficien

Re-Os and Pb-Pb geochronology of the Archean Salobo iron oxide copper-gold deposit, Carajás mineral province, northern Brazil

Rhenium-osmium ages were determined for two molybdenite samples and a Pb-Pb age was derived from bornite-chalcopyrite-magnetite at the Salobo iron oxide copper-gold deposit to determine the timing of mineralization and its relation to the nearby Old Salobo Granite. Rhenium-osmium dating of molybdenite spatially associated with copper sulfide minerals yields ages with weighted means of 2576±8 and 2562±8Ma. Removing the error multiplier introduced by the decay constant uncertainty, appropriate for comparing ages from the same isotopic system, these data convincingly argue for two temporally separated pulses of molybdenite deposition at 2576.1±1.4Ma (n=2) and 2561.7±3.1Ma (n=3). The 2576±8Ma age coincides with a previously published U-Pb age of 2573±2Ma for the Old Salobo Granite, suggesting that main stage ore formation may have been contemporaneous with granite magmatism. The slightly younger 2562Ma age most likely represents new molybdenite precipitation associated with the development or reactivation of local shear zones. Lead-lead stepwise leaching of copper sulfide minerals yields a less precise isochron age of 2579±71Ma, and supports an Archean age for the Salobo ores. This is the first documentation of an Archean iron oxide copper-gold deposit, and the Re-Os and Pb-Pb geochronology herein support 2580-2550Ma estimates for basement reactivation and regional granite magmatism associated with the development of brittle-ductile shear zone

Formation of intra-arc volcanosedimentary basins in the western flank of the central Peruvian Andes during Late Cretaceous oblique subduction: field evidence and constraints from U-Pb ages and Hf isotopes

During late Early to Late Cretaceous, the Peruvian coastal margin underwent fast and oblique subduction and was characterized by important arc plutonism (the Peruvian Coastal Batholith) and formation of volcanosedimentary basins known as the Western Peruvian Trough (WPT). We present high-precision U-Pb ages and initial Hf isotopic compositions of zircon from conformable volcanic and crosscutting intrusive rocks within submarine volcanosedimentary strata of the WPT hosting the Perubar massive sulfide deposit. Zircons extracted from both the volcanic and intrusive rocks yield concordant U-Pb ages ranging from 67.89±0.18Ma to 69.71±0.18Ma, indicating that basin subsidence, submarine volcanism and plutonic activity occurred in close spatial and temporal relationship within the Andean magmatic arc during the Late Cretaceous. Field observations, satellite image interpretation, and plate reconstructions, suggest that dextral wrenching movements along crustal lineaments were related to oblique subduction. Wrench tectonics is therefore considered to be the trigger for the formation of the WPT as a series of pull-apart basins and for the emplacement of the Coastal Batholith. The zircon initial εHf values of the dated magmatic rocks fall between 5.5 and 7.4, and indicate only very subordinate influence of a sedimentary or continental component. The absence of inherited cores in the zircons suggest a complete lack of old basement below the WPT, in agreement with previous U-Pb and Sr isotopic data for batholithic rocks emplaced in the WPT area. This is supported by the presence of a most likely continuous block of dense (~3.0g/cm3) material observed beneath the WPT area on gravimetric crustal cross sections. We suggest that this gravimetric anomaly may correspond to a piece of lithospheric mantle and/or oceanic crust inherited from a possible Late Permian-Triassic rifting. Such young and mafic crust was the most probable source for arc magmatism in the WPT are

New 40Ar/39Ar alunite ages from the Colquijirca district, Peru: evidence of a long period of magmatic SO2 degassing during formation of epithermal Au-Ag and Cordilleran polymetallic ores

We present 40Ar/39Ar data acquired by infra-red (CO2) laser step-heating of alunite crystals from the large Miocene Colquijirca district in central Peru. Combined with previously published data, our results show that a long (at least 1.3 My) and complex period of magmatic-hydrothermal activity associated with epithermal Au-(Ag) mineralization and base metal, Cordilleran ores took place at Colquijirca. The new data indicate that incursion of magmatic SO2-bearing vapor into the Colquijirca epithermal system began at least as early as ∼11.9Ma and lasted until ∼10.6Ma. Four alunite samples associated with high-sulfidation epithermal Au-(Ag) ore gave 40Ar/39Ar plateau ages between ∼11.9 and ∼11.1Ma (compared to the previously documented ∼11.6 to ∼11.3Ma). By combining individually these new ages with crosscutting relationships, the duration of the Au-(Ag) deposition period can be estimated to at least 0.4My. Three new 40Ar/39Ar plateau ages on alunite associated with the base-metal Cordilleran ores are consistent with previously obtained ages, all of them between 10.83 ± 0.06 and 10.56 ± 0.06Ma, suggesting that most of the sulfide-rich polymetallic deposits of Smelter and Colquijirca formed during this short period. The recognition of consecutive alunite-bearing and alunite-free mineral assemblages within both the Au-(Ag) and the base-metal Cordilleran ores may suggest that SO2-bearing magmatic vapor entered the epithermal environment as multiple discontinuous pulses, a number of which was not necessarily associated in time with ore fluids. It is likely that a period of SO2-bearing vapor degassing longer than 11.9 to 10.6Ma may be recognized with further more detailed wor

Caracterització i anàlisi del fenomen de les ressonàncies en sistemes elèctrics

És ben sabut que les ressonàncies en sistemes elèctrics poden elevar els valors de voltatge i corrent fins al punt d’arribar a fer malbé els circuits o, fins i tot, destruir-los. En el present treball s’estudia rigorosament el fenomen de les ressonàncies en sistemes elèctrics sota l’enfoc del mètode modal d’anàlisi, es localitzen i es quantifiquen. S’aborden en profunditat els càlculs de les sensibilitats de la freqüència de ressonància i de la impedància modal respecte als components passius no controlats clàssics dels circuits elèctrics: resistències, bobines i condensadors. Això permet saber com evolucionen les ressonàncies quan modifiquem els paràmetres dels components dels circuits. Donat que l’estabilitat dels sistemes està estretament lligada al control de les ressonàncies, l’objectiu últim és el de proporcionar les eines per poder estudiar el fenomen de l’estabilitat i controlar aquests sistemes en presència d’aquestes. Els resultats obtinguts contribueixen a una comprensió clara del fenomen i proporcionen informació pràctica respecte als circuits d’estudi. De cara a la visualització d’aquests, s’ha programat mitjançant Matlab, un codi per verificar els càlculs obtinguts i presentar de manera més didàctica els resultats. S’ha conclòs que, amb l’ajuda dels càlculs desenvolupats en aquest document i les millores obtingudes, es viable afrontar un posterior estudi de control i estabilització amb eines matemàtiques sòlides i potent